qmcintdim9

金融资产定价,随机过程,MONTE CARLO 模拟 JAVA 程序和文档资料

QUASI MONTE CARLO COMPUTATION OF TEST INTEGRALS:

We compute Int_Q F(x)dx, where Q=(0,1)^d (unit cube), d=dimension
and F(x)=h(x_1)h(x_2)...h(x_d), where h(u)=g(m*u-[m*u]).

Here [t] is the greatest integer <= t as usual. The function h(u)
is periodic with m periods on (0,1) repeating the function g(u)
m times on (0,1). If g(u) has one peak on (0,1), h(u) has m peaks
and F(x) has m^d (very narrow) peaks on Q.

DIMENSION=9


g(u)=u+0.5, periods=1, QMC relative error (%):

N: 511        SOB: -0.45    NX: -0.22     HAl: -5.06    UNF: -2.77
N: 1023       SOB: -0.29    NX: -0.36     HAl: -3.52    UNF: 1.19
N: 2047       SOB: -0.11    NX: 0.27      HAl: -2.16    UNF: 2.81
N: 4095       SOB: -0.01    NX: -0.1      HAl: -1.13    UNF: 1.56
N: 8191       SOB: 0.05     NX: -0.04     HAl: -0.7     UNF: 2.84
N: 16383      SOB: 0.04     NX: 0.0       HAl: -0.33    UNF: 0.91
N: 32767      SOB: 0.05     NX: 0.0       HAl: -0.18    UNF: 0.91



g(u)=u+0.5, periods=2, QMC relative error (%):

N: 511        SOB: -2.07    NX: -1.35     HAl: -4.12    UNF: -2.53
N: 1023       SOB: -1.69    NX: -0.85     HAl: -2.3     UNF: -2.02
N: 2047       SOB: -0.26    NX: -0.7      HAl: -1.42    UNF: -1.18
N: 4095       SOB: -0.35    NX: -0.66     HAl: -0.69    UNF: -1.57
N: 8191       SOB: -0.08    NX: -0.12     HAl: -0.3     UNF: -1.35
N: 16383      SOB: -0.07    NX: 0.0       HAl: -0.11    UNF: -1.11
N: 32767      SOB: -0.04    NX: 0.01      HAl: -0.17    UNF: -0.41



g(u)=u+0.5, periods=3, QMC relative error (%):

N: 511        SOB: -2.41    NX: 1.75      HAl: -2.29    UNF: -1.56
N: 1023       SOB: -2.27    NX: 0.44      HAl: 0.91     UNF: -0.89
N: 2047       SOB: -0.99    NX: 0.1       HAl: 0.63     UNF: 0.37
N: 4095       SOB: -0.26    NX: 0.69      HAl: 0.25     UNF: 0.19
N: 8191       SOB: 0.16     NX: 0.29      HAl: 0.13     UNF: 0.89
N: 16383      SOB: -0.08    NX: -0.02     HAl: 0.13     UNF: 1.13
N: 32767      SOB: -0.15    NX: -0.1      HAl: -0.14    UNF: 0.95



g(u)=exp(N_Inverse(u)), periods=1, QMC relative error (%):

N: 511        SOB: -18.44   NX: -58.72    HAl: -74.2    UNF: -26.36
N: 1023       SOB: -39.62   NX: -58.4     HAl: -73.81   UNF: -40.85
N: 2047       SOB: -28.67   NX: 120.2     HAl: -64.0    UNF: -34.54
N: 4095       SOB: 12.43    NX: 40.89     HAl: -57.49   UNF: -31.76
N: 8191       SOB: -6.52    NX: 16.71     HAl: -51.85   UNF: -36.87
N: 16383      SOB: -17.1    NX: -1.65     HAl: -40.94   UNF: -32.38
N: 32767      SOB: -11.19   NX: -4.67     HAl: -34.23   UNF: -18.04



g(u)=exp(N_Inverse(u)), periods=2, QMC relative error (%):

N: 511        SOB: -44.73   NX: -66.17    HAl: -100.0   UNF: -12.69
N: 1023       SOB: -34.85   NX: -55.93    HAl: -100.0   UNF: -27.31
N: 2047       SOB: -22.6    NX: -32.53    HAl: -100.0   UNF: -21.09
N: 4095       SOB: -29.59   NX: -9.72     HAl: -100.0   UNF: -32.14
N: 8191       SOB: -25.26   NX: -12.32    HAl: -100.0   UNF: -10.82
N: 16383      SOB: -8.56    NX: -2.73     HAl: -100.0   UNF: -15.61
N: 32767      SOB: -8.02    NX: -6.95     HAl: -100.0   UNF: 1.62



g(u)=exp(N_Inverse(u)), periods=3, QMC relative error (%):

N: 511        SOB: -35.92   NX: -47.4     HAl: -100.0   UNF: -11.12
N: 1023       SOB: -45.7    NX: -56.05    HAl: -100.0   UNF: -32.41
N: 2047       SOB: -39.97   NX: -30.76    HAl: -100.0   UNF: -46.0
N: 4095       SOB: -24.82   NX: -20.02    HAl: -100.0   UNF: -43.35
N: 8191       SOB: 8.65     NX: -24.8     HAl: -100.0   UNF: -34.68
N: 16383      SOB: -12.34   NX: 1.35      HAl: -100.0   UNF: 1.13
N: 32767      SOB: -14.72   NX: -8.23     HAl: -100.0   UNF: 5.06



g(u)=sin^2(2pi*u), periods=1, QMC relative error (%):

N: 511        SOB: 170.34   NX: -27.3     HAl: 5.98     UNF: 30.05
N: 1023       SOB: 83.3     NX: 7.01      HAl: -5.28    UNF: 1.38
N: 2047       SOB: 58.72    NX: 8.03      HAl: -6.3     UNF: 4.45
N: 4095       SOB: 36.7     NX: -0.16     HAl: 2.91     UNF: -1.7
N: 8191       SOB: 25.44    NX: -2.72     HAl: -2.21    UNF: -1.7
N: 16383      SOB: 3.42     NX: -0.67     HAl: -3.23    UNF: 4.96
N: 32767      SOB: 3.94     NX: -0.16     HAl: -1.18    UNF: 6.5



g(u)=sin^2(2pi*u), periods=2, QMC relative error (%):

N: 511        SOB: 393.57   NX: 5.47      HAl: -22.18   UNF: 12.13
N: 1023       SOB: 201.57   NX: 5.47      HAl: -29.86   UNF: 18.78
N: 2047       SOB: 93.54    NX: -3.23     HAl: -6.3     UNF: 4.96
N: 4095       SOB: 40.29    NX: -1.18     HAl: 2.91     UNF: -2.72
N: 8191       SOB: 25.95    NX: -1.7      HAl: 1.89     UNF: -3.23
N: 16383      SOB: 15.71    NX: 0.35      HAl: 0.86     UNF: -3.74
N: 32767      SOB: 8.54     NX: 0.86      HAl: 1.89     UNF: 2.4



g(u)=sin^2(2pi*u), periods=3, QMC relative error (%):

N: 511        SOB: 221.02   NX: 56.16     HAl: -1.7     UNF: 1.38
N: 1023       SOB: 101.73   NX: 21.34     HAl: -8.86    UNF: -3.23
N: 2047       SOB: 62.3     NX: 10.08     HAl: 6.5      UNF: -8.35
N: 4095       SOB: 29.54    NX: 3.94      HAl: 5.47     UNF: 0.35
N: 8191       SOB: 12.13    NX: -2.72     HAl: 2.91     UNF: -4.77
N: 16383      SOB: 10.59    NX: -1.18     HAl: 1.38     UNF: 0.86
N: 32767      SOB: 8.54     NX: 0.35      HAl: 1.38     UNF: -0.67



g(u)=3u^2, periods=1, QMC relative error (%):

N: 511        SOB: -18.03   NX: -28.19    HAl: -43.63   UNF: -34.61
N: 1023       SOB: -28.55   NX: -35.54    HAl: -49.41   UNF: -42.43
N: 2047       SOB: -22.9    NX: 88.87     HAl: -33.04   UNF: -40.68
N: 4095       SOB: 26.94    NX: 34.24     HAl: -27.31   UNF: 4.3
N: 8191       SOB: 17.79    NX: 12.16     HAl: -24.13   UNF: 5.87
N: 16383      SOB: 9.85     NX: 0.55      HAl: -7.73    UNF: -0.03
N: 32767      SOB: 7.77     NX: -0.55     HAl: -4.8     UNF: 5.43



g(u)=4u^3, periods=2, QMC relative error (%):

N: 511        SOB: -50.68   NX: -72.13    HAl: -45.84   UNF: 6.15
N: 1023       SOB: -54.82   NX: -60.06    HAl: 21.12    UNF: 65.49
N: 2047       SOB: -25.24   NX: -21.33    HAl: -16.39   UNF: -6.96
N: 4095       SOB: -45.48   NX: -26.46    HAl: -34.72   UNF: -30.14
N: 8191       SOB: -24.53   NX: -33.01    HAl: -26.02   UNF: -38.4
N: 16383      SOB: -26.21   NX: 2.56      HAl: -25.16   UNF: -16.51
N: 32767      SOB: -23.75   NX: -5.1      HAl: -20.75   UNF: 3.82



g(u)=5u^4, periods=3, QMC relative error (%):

N: 511        SOB: 283.07   NX: -79.38    HAl: -90.02   UNF: -70.66
N: 1023       SOB: 104.72   NX: -87.75    HAl: -54.03   UNF: -78.31
N: 2047       SOB: 9.28     NX: -46.9     HAl: -46.93   UNF: -76.05
N: 4095       SOB: -38.81   NX: -3.77     HAl: -50.73   UNF: -59.4
N: 8191       SOB: 47.76    NX: -38.89    HAl: -23.25   UNF: -70.22
N: 16383      SOB: -13.88   NX: -42.27    HAl: -20.52   UNF: -69.06
N: 32767      SOB: -11.63   NX: -39.28    HAl: -40.97   UNF: -57.59



g(u)=2-12*(u-0.5)^2, periods=1, QMC relative error (%):

N: 511        SOB: -71.44   NX: -20.18    HAl: -39.67   UNF: 35.01
N: 1023       SOB: -14.36   NX: -15.41    HAl: 21.15    UNF: 17.01
N: 2047       SOB: -0.2     NX: -31.56    HAl: 22.23    UNF: -5.2
N: 4095       SOB: -15.79   NX: -33.29    HAl: -12.43   UNF: -0.64
N: 8191       SOB: -26.71   NX: -18.62    HAl: -10.7    UNF: 11.43
N: 16383      SOB: -15.6    NX: -12.39    HAl: -15.1    UNF: 1.66
N: 32767      SOB: -5.44    NX: -4.48     HAl: -4.8     UNF: 3.95



g(u)=2-12*(u-0.5)^2, periods=2, QMC relative error (%):

N: 511        SOB: 94.34    NX: -74.17    HAl: 36.44    UNF: -85.31
N: 1023       SOB: 65.72    NX: 23.1      HAl: 23.87    UNF: -64.08
N: 2047       SOB: 52.45    NX: 29.15     HAl: -17.05   UNF: -20.08
N: 4095       SOB: 45.33    NX: -5.37     HAl: 9.12     UNF: -12.66
N: 8191       SOB: 40.58    NX: -15.05    HAl: -8.28    UNF: 3.23
N: 16383      SOB: -3.65    NX: -4.36     HAl: -7.07    UNF: 5.38
N: 32767      SOB: 5.92     NX: -4.74     HAl: -6.35    UNF: 5.63



g(u)=2-12*(u-0.5)^2, periods=3, QMC relative error (%):

N: 511        SOB: 29.54    NX: 10.93     HAl: 54.22    UNF: 115.9
N: 1023       SOB: -1.73    NX: -27.69    HAl: -22.7    UNF: 74.89
N: 2047       SOB: -3.79    NX: -42.99    HAl: -28.24   UNF: 65.04
N: 4095       SOB: -12.08   NX: -44.92    HAl: -5.99    UNF: 23.97
N: 8191       SOB: -11.06   NX: -20.12    HAl: -6.49    UNF: -1.99
N: 16383      SOB: 5.27     NX: 3.71      HAl: -0.91    UNF: 4.23
N: 32767      SOB: 9.07     NX: 0.61      HAl: -3.47    UNF: 2.73



g(u)=3*I_[1/3,2/3](u), periods=1, QMC relative error (%):

N: 511        SOB: -100.0   NX: -100.0    HAl: -100.0   UNF: -100.0
N: 1023       SOB: -100.0   NX: -100.0    HAl: -100.0   UNF: -100.0
N: 2047       SOB: -100.0   NX: -100.0    HAl: -100.0   UNF: -100.0
N: 4095       SOB: -100.0   NX: -100.0    HAl: -100.0   UNF: -100.0
N: 8191       SOB: 140.33   NX: -100.0    HAl: -100.0   UNF: 140.33
N: 16383      SOB: 20.15    NX: -100.0    HAl: -100.0   UNF: 20.15
N: 32767      SOB: 20.14    NX: -100.0    HAl: -39.93   UNF: 20.14



g(u)=3*I_[1/3,2/3](u), periods=2, QMC relative error (%):

N: 511        SOB: 7618.82  NX: -100.0    HAl: -100.0   UNF: -100.0
N: 1023       SOB: 3751.86  NX: -100.0    HAl: -100.0   UNF: -100.0
N: 2047       SOB: 1824.05  NX: -100.0    HAl: -100.0   UNF: -100.0
N: 4095       SOB: 861.55   NX: -100.0    HAl: -100.0   UNF: -100.0
N: 8191       SOB: 380.66   NX: -100.0    HAl: -100.0   UNF: -100.0
N: 16383      SOB: 140.3    NX: 140.3     HAl: -100.0   UNF: 20.15
N: 32767      SOB: 140.29   NX: 20.14     HAl: 20.14    UNF: 140.29



g(u)=3*I_[1/3,2/3](u), periods=3, QMC relative error (%):

N: 511        SOB: -100.0   NX: -100.0    HAl: -100.0   UNF: -100.0
N: 1023       SOB: -100.0   NX: -100.0    HAl: -100.0   UNF: -100.0
N: 2047       SOB: -100.0   NX: -100.0    HAl: -100.0   UNF: -100.0
N: 4095       SOB: -100.0   NX: -100.0    HAl: -100.0   UNF: -100.0
N: 8191       SOB: -100.0   NX: -100.0    HAl: 140.33   UNF: -100.0
N: 16383      SOB: 20.15    NX: 20.15     HAl: 20.15    UNF: 140.3
N: 32767      SOB: 140.29   NX: 80.21     HAl: -39.93   UNF: 80.21